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(* Title: HoTT/Nat.thy
Author: Josh Chen
Date: Aug 2018
Natural numbers.
*)
theory Nat
imports HoTT_Base
begin
section ‹Constants and type rules›
axiomatization
Nat :: Term ("ℕ") and
zero :: Term ("0") and
succ :: "Term ⇒ Term" and
indNat :: "[[Term, Term] ⇒ Term, Term, Term] ⇒ Term" ("(1ind⇩ℕ)")
where
Nat_form: "ℕ: U(O)"
and
Nat_intro1: "0: ℕ"
and
Nat_intro2: "n: ℕ ⟹ succ(n): ℕ"
and
Nat_elim: "⟦
C: ℕ ⟶ U(i);
⋀n c. ⟦n: ℕ; c: C(n)⟧ ⟹ f(n)(c): C(succ n);
a: C(0);
n: ℕ
⟧ ⟹ ind⇩ℕ(f)(a)(n): C(n)"
and
Nat_comp1: "⟦
C: ℕ ⟶ U(i);
⋀n c. ⟦n: ℕ; c: C(n)⟧ ⟹ f(n)(c): C(succ n);
a: C(0)
⟧ ⟹ ind⇩ℕ(f)(a)(0) ≡ a"
and
Nat_comp2: "⟦
C: ℕ ⟶ U(i);
⋀n c. ⟦n: ℕ; c: C(n)⟧ ⟹ f(n)(c): C(succ n);
a: C(0);
n: ℕ
⟧ ⟹ ind⇩ℕ(f)(a)(succ n) ≡ f(n)(ind⇩ℕ f a n)"
text "Rule declarations:"
lemmas Nat_intro = Nat_intro1 Nat_intro2
lemmas Nat_rules [intro] = Nat_form Nat_intro Nat_elim Nat_comp1 Nat_comp2
lemmas Nat_comps [comp] = Nat_comp1 Nat_comp2
end
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